To determine the average rate of change of the function $f(x) = x^2 - 2x + 3$ from $x = 2$ to $x = 7$, we can use the formula:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

where $a = 2$ and $b = 7$.

Step-by-Step Solution

1. Calculate $f(2)$: $f(2) = (2)^2 - 2(2) + 3 = 4 - 4 + 3 = 3$

2. Calculate $f(7)$: $f(7) = (7)^2 - 2(7) + 3 = 49 - 14 + 3 = 38$

3. Apply the average rate of change formula: $\text{Average rate of change} = \frac{f(7) - f(2)}{7 - 2} = \frac{38 - 3}{5} = \frac{35}{5} = 7$

Answer

The average rate of change of $f(x)$ from $x = 2$ to $x = 7$ is 7.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How does the average rate of change differ from the instantaneous rate of change?
2. What would the average rate of change be from $x = 3$ to $x = 6$?
3. How would the result change if we used a different quadratic function?
4. Can the average rate of change be negative? If so, in what scenario?
5. How do we interpret the result of an average rate of change in terms of the graph of the function?

Tip: For a quadratic function $f(x) = ax^2 + bx + c$, the average rate of change over any interval is affected by the shape (parabolic curvature) and the endpoints chosen.